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We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr{o}dinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization coordinate variable in order to avoid multi-value problems. Based on a new complex plane, the direct scattering problem perform a detailed analysis of the analytical, asymptotic and symmetry properties of the Jost functions and the scattering matrix. Then, a generalized Riemann-Hilbert problem (RHP) is successfully established from the results of the direct scattering transform. In the inverse scattering problem, we discuss the discrete spectrum, residue condition, trace formula and theta condition under simple poles and double poles respectively, and further solve the solution of a generalized RHP. Finally, we derive the solution of the equation for the cases of different poles without reflection potential. In addition, we analyze the localized structures and dynamic behaviors of the resulting soliton solutions by taking some appropriate values of the parameters appeared in the solutions.
The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schr{o}dinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residue
In this work, we consider the generalized variable-coefficient nonlinear Schr{o}dinger equation with non-vanishing boundary conditions at infinity including the simple and double poles of the scattering coefficients. By introducing an appropriate Rie
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asy
We derive a straightforward variational method to construct embedded soliton solutions of the third-order nonlinear Schodinger equation and analytically demonstrate that these solitons exist as a continuous family. We argue that a particular embedded
We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on